The Fate of Singularities in Quantum Cosmology and the Application of Generalized Effective Equations to Constrained Quantum Systems

The first part of the thesis is concerned with the fate of singularities in quantum cosmology. The second part addresses the derivation of predictions from quantum cosmology. In the first part, two classes of cosmological models were studied. In the first class of models, the universe evolves to or emerges from a big-rip singularity. Here, energy density, pressure and scale factor diverge after a finite amount of time. This type of singularity arises rather generically in cosmological models with phantom dark energy. For each of these phantom-field models, the corresponding scenario with ordinary scalar field was studied. The scalar field induced a big-bang singularity. The second class of models studied was dominated by a big-brake singularity. At the big brake, the universe evolution comes to a halt due to an infinite deceleration. The motivation behind this choice of models was the occurrence of a singularity at large scale factor. The major question pursued was whether these types of singularity were resolved on the quantum level. If such singularities were resolved in quantum cosmology, this would imply that quantum gravitational effects can occur in the macroscopic universe. After devising classical models that contain the respective singularity, I subjected these models to quantization which was carried out in the geometrodynamical approach. The governing equation is then the Wheeler�DeWitt equation. I found solutions to the Wheeler�DeWitt equation, in one case even an exact solution. Wave packets were constructed around trajectories which, on the classical level, would lead into the singularity. I have then shown that the classical trajectory can indeed be recovered from these packets through the principle of constructive interference. As criteria for singularity avoidance the vanishing of the wave function at the location of the classical singularity, as well as the spreading of wave packets upon approach of this region was used. Whereas the former ensures that the classical singularity does not contribute to the quantum theory, the latter signals a dissolution of the semi-classical approximation and thus of spacetime. In all cases, I found singularity resolution. In the case of the big-bang and big-brake singularities, the wave function vanishes at the classical singularity. These two have in common that they occur at finite value of the configuration space variables. A spreading of the wave packet is however only observed upon approach of the big-brake singularity. At the location of the classical big-rip singularity, a strict vanishing cannot be found. This singularity is located at the infinite boundary of configuration space. The wave packet spreads upon approach of this singularity. The second part of my thesis deals with the application of the generalized effective-equation scheme to constrained systems. Generalized effective equations describe a quantum system via expectation values of fundamental operators and higher moments of the wave function � instead of using the wave function itself. It is thus a very useful scheme for the derivation of predictions from quantum cosmology, e.g. in the form of corrections to classical equations of motion. The theory is formulated on a, generally, infinite-dimensional quantum phase space. The first task was to find a for- mulation of Dirac�s constraint-quantization condition on this phase space. Such a formulation was found and proven to remove degrees of freedom appropriately in the case of a single linear constraint. This result ensures the correct removal of degrees of freedom for any singly constrained system at least locally. In a second step, the newly formulated constraints � there are actually infinitely many of them � had to be consistently approximated. Such an approximation is necessary to reduce the infinite number of con- straints to a finite one. Only then can information be extracted from the system. Such an approximation scheme for non-relativistic systems was developed. Its consistency was explicitly checked using the parametrised, free non-relativistic particle

Effective Constraints for Quantum Systems

An effective formalism for quantum constrained systems is presented which allows manageable derivations of solutions and observables, including a treatment of physical reality conditions without requiring full knowledge of the physical inner product. Instead of a state equation from a constraint operator, an infinite system of constraint functions on the quantum phase space of expectation values and moments of states is used. The examples of linear constraints as well as the free non-relativistic particle in parameterized form illustrate how standard problems of constrained systems can be dealt with in this framework.Comment: 40 page

Quantum cosmology with big-brake singularity

We investigate a cosmological model with a big-brake singularity in the future: while the first time derivative of the scale factor goes to zero, its second time derivative tends to minus infinity. Although we also discuss the classical version of the model in some detail, our main interest lies in its quantization. We formulate the Wheeler-DeWitt equation and derive solutions describing wave packets. We show that all such solutions vanish in the region of the classical singularity, a behaviour which we interpret as singularity avoidance. We then discuss the same situation in loop quantum cosmology. While this leads to a different factor ordering, the singularity is there avoided, too.Comment: 24 pages, 7 figures, figures improved, references added, conceptual clarifications include

Quantum phantom cosmology

We apply the formalism of quantum cosmology to models containing a phantom field. Three models are discussed explicitly: a toy model, a model with an exponential phantom potential, and a model with phantom field accompanied by a negative cosmological constant. In all these cases we calculate the classical trajectories in configuration space and give solutions to the Wheeler-DeWitt equation in quantum cosmology. In the cases of the toy model and the model with exponential potential we are able to solve the Wheeler-DeWitt equation exactly. For comparison, we also give the corresponding solutions for an ordinary scalar field. We discuss in particular the behaviour of wave packets in minisuperspace. For the phantom field these packets disperse in the region that corresponds to the Big Rip singularity. This thus constitutes a genuine quantum region at large scales, described by a regular solution of the Wheeler-DeWitt equation. For the ordinary scalar field, the Big-Bang singularity is avoided. Some remarks on the arrow of time in phantom models as well as on the relation of phantom models to loop quantum cosmology are given.Comment: 21 pages, 6 figure